“Spectral imaging” is commonly understood as referring to imaging with a limited number of wavelengths (e.g. up to 10) in a given wavelength range, e.g. the visible (“VIS”) range or the near-infrared (“NIR”) range. “Hyperspectral imaging” is commonly understood as referring to imaging with a larger number of wavelengths in a given wavelength range, for example between 10 and hundreds or even thousands of wavelengths. “Snapshot” SI and HSI imagers perform simultaneous (instantaneous) acquisition of spatial and spectral data in a single snapshot. The data acquired forms a “spatial-spectral cube” (also referred to herein simply as “spectral cube” or “data cube”) of a source object (also referred to simply as “object” or “scene”). “Spatial-spectral cube”, “spectral cube” and “data cube” are hereinafter used interchangeably. A data cube includes light intensity data in two spatial dimensions and one spectral dimension and is expressed as a three-dimensional (3D) matrix.
Commonly authored and assigned U.S. patent application Ser. No. 13/752,560 titled “Snapshot spectral imaging based on digital cameras” (published as US Pat. Pub. 20130194481), which is incorporated herein by reference in its entirety, teaches compressed sensing (CS)-based snapshot spectral imaging (CS-SSI) in apparatus including an imaging lens, a dispersed image sensor and a restricted isometry property (RIP) diffuser inserted in the optical path between a source image and a pixelated (as. e.g. in a digital camera) image sensor. The RIP diffuser may be one dimensional (1D). It provides a dispersed and diffused image (“DD image”) at the dispersed image sensor. Due to the 1D RIP diffuser optical properties, each pixel in the DD image includes a linear mixture of spectral and spatial information from all pixels of a corresponding column in the DD image. In US 20130194481, full reconstruction of the data cube is performed using a CS-based optimization process to compensate for the underdetermined nature of the problem. The operator performing the linear projections can be described as a “sensing matrix” that has fewer rows than columns and that operates on the data cube to form a DD image. The reconstruction process guarantees full reconstruction of the source object if the sensing matrix satisfies a RIP condition. The RIP diffuser is designed such that the transfer-function (which is identical with the sensing matrix) of an optical imaging system including the diffuser satisfies the RIP condition at each single wavelength (or at a band chosen around a single wavelength).
The solution provided in US 20130194481 performs 1D CS-SCR using block Toeplitz matrices to perform a single 1D transform applied sequentially to columns of an array that comprises all the wavebands images concatenated in a vertical direction. It has been shown that the RIP condition for block Toeplitz matrices is harder to uphold than for random ones, in terms of the sparsity required from the signal one wishes to reconstruct.